step-75.h

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) const override
 *       {
 *         const std::array<double, 2> polar =
 *           GeometricUtilities::Coordinates::to_spherical(Point<2>(p[0], p[1]));
 *   
 *         constexpr const double alpha = 2. / 3.;
 *         return std::pow(polar[0], alpha) * std::sin(alpha * polar[1]);
 *       }
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-Parameters"></a> 
 * <h3>Parameters</h3>
 * 
 
 * 
 * For this tutorial, we will use a simplified set of parameters. It is also
 * possible to use a ParameterHandler class here, but to keep this tutorial
 * short we decided on using simple structs. The actual intention of all these
 * parameters will be described in the upcoming classes at their respective
 * location where they are used.
 *   
 
 * 
 * The following parameter set controls the coarse-grid solver, the smoothers,
 * and the inter-grid transfer scheme of the multigrid mechanism.
 * We populate it with default parameters.
 * 
 * @code
 *     struct MultigridParameters
 *     {
 *       struct
 *       {
 *         std::string  type            = "cg_with_amg";
 *         unsigned int maxiter         = 10000;
 *         double       abstol          = 1e-20;
 *         double       reltol          = 1e-4;
 *         unsigned int smoother_sweeps = 1;
 *         unsigned int n_cycles        = 1;
 *         std::string  smoother_type   = "ILU";
 *       } coarse_solver;
 *   
 *       struct
 *       {
 *         std::string  type                = "chebyshev";
 *         double       smoothing_range     = 20;
 *         unsigned int degree              = 5;
 *         unsigned int eig_cg_n_iterations = 20;
 *       } smoother;
 *   
 *       struct
 *       {
 *         MGTransferGlobalCoarseningTools::PolynomialCoarseningSequenceType
 *           p_sequence = MGTransferGlobalCoarseningTools::
 *             PolynomialCoarseningSequenceType::decrease_by_one;
 *         bool perform_h_transfer = true;
 *       } transfer;
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * This is the general parameter struct for the problem class. You will find
 * this struct divided into several categories, including general runtime
 * parameters, level limits, refine and coarsen fractions, as well as
 * parameters for cell weighting. It also contains an instance of the above
 * struct for multigrid parameters which will be passed to the multigrid
 * algorithm.
 * 
 * @code
 *     struct Parameters
 *     {
 *       unsigned int n_cycles         = 8;
 *       double       tolerance_factor = 1e-12;
 *   
 *       MultigridParameters mg_data;
 *   
 *       unsigned int min_h_level            = 5;
 *       unsigned int max_h_level            = 12;
 *       unsigned int min_p_degree           = 2;
 *       unsigned int max_p_degree           = 6;
 *       unsigned int max_p_level_difference = 1;
 *   
 *       double refine_fraction    = 0.3;
 *       double coarsen_fraction   = 0.03;
 *       double p_refine_fraction  = 0.9;
 *       double p_coarsen_fraction = 0.9;
 *   
 *       double weighting_factor   = 1.;
 *       double weighting_exponent = 1.;
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-MatrixfreeLaplaceoperator"></a> 
 * <h3>Matrix-free Laplace operator</h3>
 * 
 
 * 
 * This is a matrix-free implementation of the Laplace operator that will
 * basically take over the part of the `assemble_system()` function from other
 * tutorials. The meaning of all member functions will be explained at their
 * definition later.
 *   
 
 * 
 * We will use the FEEvaluation class to evaluate the solution vector
 * at the quadrature points and to perform the integration. In contrast to
 * other tutorials, the template arguments `degree` is set to @f$-1@f$ and
 * `number of quadrature in 1d` to @f$0@f$. In this case, FEEvaluation selects
 * dynamically the correct polynomial degree and number of quadrature
 * points. Here, we introduce an alias to FEEvaluation with the correct
 * template parameters so that we do not have to worry about them later on.
 * 
 * @code
 *     template <int dim, typename number>
 *     class LaplaceOperator : public EnableObserverPointer
 *     {
 *     public:
 *       using VectorType = LinearAlgebra::distributed::Vector<number>;
 *   
 *       using FECellIntegrator = FEEvaluation<dim, -1, 0, 1, number>;
 *   
 *       LaplaceOperator() = default;
 *   
 *       LaplaceOperator(const hp::MappingCollection<dim> &mapping,
 *                       const DoFHandler<dim>            &dof_handler,
 *                       const hp::QCollection<dim>       &quad,
 *                       const AffineConstraints<number>  &constraints,
 *                       VectorType                       &system_rhs);
 *   
 *       void reinit(const hp::MappingCollection<dim> &mapping,
 *                   const DoFHandler<dim>            &dof_handler,
 *                   const hp::QCollection<dim>       &quad,
 *                   const AffineConstraints<number>  &constraints,
 *                   VectorType                       &system_rhs);
 *   
 *       types::global_dof_index m() const;
 *   
 *       number el(unsigned int, unsigned int) const;
 *   
 *       void initialize_dof_vector(VectorType &vec) const;
 *   
 *       void vmult(VectorType &dst, const VectorType &src) const;
 *   
 *       void Tvmult(VectorType &dst, const VectorType &src) const;
 *   
 *       const TrilinosWrappers::SparseMatrix &get_system_matrix() const;
 *   
 *       void compute_inverse_diagonal(VectorType &diagonal) const;
 *   
 *     private:
 *       void do_cell_integral_local(FECellIntegrator &integrator) const;
 *   
 *       void do_cell_integral_global(FECellIntegrator &integrator,
 *                                    VectorType       &dst,
 *                                    const VectorType &src) const;
 *   
 *   
 *       void do_cell_integral_range(
 *         const MatrixFree<dim, number>               &matrix_free,
 *         VectorType                                  &dst,
 *         const VectorType                            &src,
 *         const std::pair<unsigned int, unsigned int> &range) const;
 *   
 *       MatrixFree<dim, number> matrix_free;
 *   
 * @endcode
 * 
 * To solve the equation system on the coarsest level with an AMG
 * preconditioner, we need an actual system matrix on the coarsest level.
 * For this purpose, we provide a mechanism that optionally computes a
 * matrix from the matrix-free formulation, for which we introduce a
 * dedicated SparseMatrix object. In the default case, this matrix stays
 * empty. Once `get_system_matrix()` is called, this matrix is filled (lazy
 * allocation). Since this is a `const` function, we need the "mutable"
 * keyword here. We also need a the constraints object to build the matrix.
 * 
 * @code
 *       AffineConstraints<number>              constraints;
 *       mutable TrilinosWrappers::SparseMatrix system_matrix;
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * The following section contains functions to initialize and reinitialize
 * the class. In particular, these functions initialize the internal
 * MatrixFree instance. For sake of simplicity, we also compute the system
 * right-hand-side vector.
 * 
 * @code
 *     template <int dim, typename number>
 *     LaplaceOperator<dim, number>::LaplaceOperator(
 *       const hp::MappingCollection<dim> &mapping,
 *       const DoFHandler<dim>            &dof_handler,
 *       const hp::QCollection<dim>       &quad,
 *       const AffineConstraints<number>  &constraints,
 *       VectorType                       &system_rhs)
 *     {
 *       this->reinit(mapping, dof_handler, quad, constraints, system_rhs);
 *     }
 *   
 *   
 *   
 *     template <int dim, typename number>
 *     void LaplaceOperator<dim, number>::reinit(
 *       const hp::MappingCollection<dim> &mapping,
 *       const DoFHandler<dim>            &dof_handler,
 *       const hp::QCollection<dim>       &quad,
 *       const AffineConstraints<number>  &constraints,
 *       VectorType                       &system_rhs)
 *     {
 * @endcode
 * 
 * Clear internal data structures (in the case that the operator is reused).
 * 
 * @code
 *       this->system_matrix.clear();
 *   
 * @endcode
 * 
 * Copy the constraints, since they might be needed for computation of the
 * system matrix later on.
 * 
 * @code
 *       this->constraints.copy_from(constraints);
 *   
 * @endcode
 * 
 * Set up MatrixFree. At the quadrature points, we only need to evaluate
 * the gradient of the solution and test with the gradient of the shape
 * functions so that we only need to set the flag `update_gradients`.
 * 
 * @code
 *       typename MatrixFree<dim, number>::AdditionalData data;
 *       data.mapping_update_flags = update_gradients;
 *   
 *       matrix_free.reinit(mapping, dof_handler, constraints, quad, data);
 *   
 * @endcode
 * 
 * Compute the right-hand side vector. For this purpose, we set up a second
 * MatrixFree instance that uses a modified AffineConstraints not containing
 * the constraints due to Dirichlet-boundary conditions. This modified
 * operator is applied to a vector with only the Dirichlet values set. The
 * result is the negative right-hand-side vector.
 * 
 * @code
 *       {
 *         AffineConstraints<number> constraints_without_dbc(
 *           dof_handler.locally_owned_dofs(),
 *           DoFTools::extract_locally_relevant_dofs(dof_handler));
 *   
 *         DoFTools::make_hanging_node_constraints(dof_handler,
 *                                                 constraints_without_dbc);
 *         constraints_without_dbc.close();
 *   
 *         VectorType b, x;
 *   
 *         this->initialize_dof_vector(system_rhs);
 *   
 *         MatrixFree<dim, number> matrix_free;
 *         matrix_free.reinit(
 *           mapping, dof_handler, constraints_without_dbc, quad, data);
 *   
 *         matrix_free.initialize_dof_vector(b);
 *         matrix_free.initialize_dof_vector(x);
 *   
 *         constraints.distribute(x);
 *   
 *         matrix_free.cell_loop(&LaplaceOperator::do_cell_integral_range,
 *                               this,
 *                               b,
 *                               x);
 *   
 *         constraints.set_zero(b);
 *   
 *         system_rhs -= b;
 *       }
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * The following functions are implicitly needed by the multigrid algorithm,
 * including the smoothers.
 * 
 
 * 
 * Since we do not have a matrix, query the DoFHandler for the number of
 * degrees of freedom.
 * 
 * @code
 *     template <int dim, typename number>
 *     types::global_dof_index LaplaceOperator<dim, number>::m() const
 *     {
 *       return matrix_free.get_dof_handler().n_dofs();
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * Access a particular element in the matrix. This function is neither
 * needed nor implemented, however, is required to compile the program.
 * 
 * @code
 *     template <int dim, typename number>
 *     number LaplaceOperator<dim, number>::el(unsigned int, unsigned int) const
 *     {
 *       DEAL_II_NOT_IMPLEMENTED();
 *       return 0;
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * Initialize the given vector. We simply delegate the task to the
 * MatrixFree function with the same name.
 * 
 * @code
 *     template <int dim, typename number>
 *     void
 *     LaplaceOperator<dim, number>::initialize_dof_vector(VectorType &vec) const
 *     {
 *       matrix_free.initialize_dof_vector(vec);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * Perform an operator evaluation by looping with the help of MatrixFree
 * over all cells and evaluating the effect of the cell integrals (see also:
 * `do_cell_integral_local()` and `do_cell_integral_global()`).
 * 
 * @code
 *     template <int dim, typename number>
 *     void LaplaceOperator<dim, number>::vmult(VectorType       &dst,
 *                                              const VectorType &src) const
 *     {
 *       this->matrix_free.cell_loop(
 *         &LaplaceOperator::do_cell_integral_range, this, dst, src, true);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * Perform the transposed operator evaluation. Since we are considering
 * symmetric "matrices", this function can simply delegate it task to vmult().
 * 
 * @code
 *     template <int dim, typename number>
 *     void LaplaceOperator<dim, number>::Tvmult(VectorType       &dst,
 *                                               const VectorType &src) const
 *     {
 *       this->vmult(dst, src);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * Since we do not have a system matrix, we cannot loop over the the
 * diagonal entries of the matrix. Instead, we compute the diagonal by
 * performing a sequence of operator evaluations to unit basis vectors.
 * For this purpose, an optimized function from the MatrixFreeTools
 * namespace is used. The inversion is performed manually afterwards.
 * 
 * @code
 *     template <int dim, typename number>
 *     void LaplaceOperator<dim, number>::compute_inverse_diagonal(
 *       VectorType &diagonal) const
 *     {
 *       this->matrix_free.initialize_dof_vector(diagonal);
 *       MatrixFreeTools::compute_diagonal(matrix_free,
 *                                         diagonal,
 *                                         &LaplaceOperator::do_cell_integral_local,
 *                                         this);
 *   
 *       for (auto &i : diagonal)
 *         i = (std::abs(i) > 1.0e-10) ? (1.0 / i) : 1.0;
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * In the matrix-free context, no system matrix is set up during
 * initialization of this class. As a consequence, it has to be computed
 * here if it should be requested. Since the matrix is only computed in
 * this tutorial for linear elements (on the coarse grid), this is
 * acceptable.
 * The matrix entries are obtained via sequence of operator evaluations.
 * For this purpose, the optimized function MatrixFreeTools::compute_matrix()
 * is used. The matrix will only be computed if it has not been set up yet
 * (lazy allocation).
 * 
 * @code
 *     template <int dim, typename number>
 *     const TrilinosWrappers::SparseMatrix &
 *     LaplaceOperator<dim, number>::get_system_matrix() const
 *     {
 *       if (system_matrix.m() == 0 && system_matrix.n() == 0)
 *         {
 *           const auto &dof_handler = this->matrix_free.get_dof_handler();
 *   
 *           const IndexSet locally_relevant_dofs =
 *             DoFTools::extract_locally_relevant_dofs(dof_handler);
 *           DynamicSparsityPattern dsp(locally_relevant_dofs);
 *           DoFTools::make_sparsity_pattern(dof_handler, dsp, this->constraints);
 *   
 *           SparsityTools::distribute_sparsity_pattern(
 *             dsp,
 *             dof_handler.locally_owned_dofs(),
 *             dof_handler.get_triangulation().get_mpi_communicator(),
 *             locally_relevant_dofs);
 *   
 *           system_matrix.reinit(dsp);
 *   
 *           MatrixFreeTools::compute_matrix(
 *             matrix_free,
 *             constraints,
 *             system_matrix,
 *             &LaplaceOperator::do_cell_integral_local,
 *             this);
 *         }
 *   
 *       return this->system_matrix;
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * Perform cell integral on a cell batch without gathering and scattering
 * the values. This function is needed for the MatrixFreeTools functions
 * since these functions operate directly on the buffers of FEEvaluation.
 * 
 * @code
 *     template <int dim, typename number>
 *     void LaplaceOperator<dim, number>::do_cell_integral_local(
 *       FECellIntegrator &integrator) const
 *     {
 *       integrator.evaluate(EvaluationFlags::gradients);
 *   
 *       for (const unsigned int q : integrator.quadrature_point_indices())
 *         integrator.submit_gradient(integrator.get_gradient(q), q);
 *   
 *       integrator.integrate(EvaluationFlags::gradients);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * Same as above but with access to the global vectors.
 * 
 * @code
 *     template <int dim, typename number>
 *     void LaplaceOperator<dim, number>::do_cell_integral_global(
 *       FECellIntegrator &integrator,
 *       VectorType       &dst,
 *       const VectorType &src) const
 *     {
 *       integrator.gather_evaluate(src, EvaluationFlags::gradients);
 *   
 *       for (const unsigned int q : integrator.quadrature_point_indices())
 *         integrator.submit_gradient(integrator.get_gradient(q), q);
 *   
 *       integrator.integrate_scatter(EvaluationFlags::gradients, dst);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * This function loops over all cell batches within a cell-batch range and
 * calls the above function.
 * 
 * @code
 *     template <int dim, typename number>
 *     void LaplaceOperator<dim, number>::do_cell_integral_range(
 *       const MatrixFree<dim, number>               &matrix_free,
 *       VectorType                                  &dst,
 *       const VectorType                            &src,
 *       const std::pair<unsigned int, unsigned int> &range) const
 *     {
 *       FECellIntegrator integrator(matrix_free, range);
 *   
 *       for (unsigned cell = range.first; cell < range.second; ++cell)
 *         {
 *           integrator.reinit(cell);
 *   
 *           do_cell_integral_global(integrator, dst, src);
 *         }
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-Solverandpreconditioner"></a> 
 * <h3>Solver and preconditioner</h3>
 * 
 
 * 
 * 
 * <a name="step_75-Conjugategradientsolverwithmultigridpreconditioner"></a> 
 * <h4>Conjugate-gradient solver with multigrid preconditioner</h4>
 * 
 
 * 
 * This function solves the equation system with a sequence of provided
 * multigrid objects. It is meant to be treated as general as possible, hence
 * the multitude of template parameters.
 * 
 * @code
 *     template <typename VectorType,
 *               int dim,
 *               typename SystemMatrixType,
 *               typename LevelMatrixType,
 *               typename MGTransferType>
 *     static void
 *     mg_solve(SolverControl             &solver_control,
 *              VectorType                &dst,
 *              const VectorType          &src,
 *              const MultigridParameters &mg_data,
 *              const DoFHandler<dim>     &dof,
 *              const SystemMatrixType    &fine_matrix,
 *              const MGLevelObject<std::unique_ptr<LevelMatrixType>> &mg_matrices,
 *              const MGTransferType                                  &mg_transfer)
 *     {
 *       AssertThrow(mg_data.coarse_solver.type == "cg_with_amg",
 *                   ExcNotImplemented());
 *       AssertThrow(mg_data.smoother.type == "chebyshev", ExcNotImplemented());
 *   
 *       const unsigned int min_level = mg_matrices.min_level();
 *       const unsigned int max_level = mg_matrices.max_level();
 *   
 *       using SmootherPreconditionerType = DiagonalMatrix<VectorType>;
 *       using SmootherType               = PreconditionChebyshev<LevelMatrixType,
 *                                                  VectorType,
 *                                                  SmootherPreconditionerType>;
 *       using PreconditionerType = PreconditionMG<dim, VectorType, MGTransferType>;
 *   
 * @endcode
 * 
 * We initialize level operators and Chebyshev smoothers here.
 * 
 * @code
 *       mg::Matrix<VectorType> mg_matrix(mg_matrices);
 *   
 *       MGLevelObject<typename SmootherType::AdditionalData> smoother_data(
 *         min_level, max_level);
 *   
 *       for (unsigned int level = min_level; level <= max_level; ++level)
 *         {
 *           smoother_data[level].preconditioner =
 *             std::make_shared<SmootherPreconditionerType>();
 *           mg_matrices[level]->compute_inverse_diagonal(
 *             smoother_data[level].preconditioner->get_vector());
 *           smoother_data[level].smoothing_range = mg_data.smoother.smoothing_range;
 *           smoother_data[level].degree          = mg_data.smoother.degree;
 *           smoother_data[level].eig_cg_n_iterations =
 *             mg_data.smoother.eig_cg_n_iterations;
 *         }
 *   
 *       MGSmootherPrecondition<LevelMatrixType, SmootherType, VectorType>
 *         mg_smoother;
 *       mg_smoother.initialize(mg_matrices, smoother_data);
 *   
 * @endcode
 * 
 * Next, we initialize the coarse-grid solver. We use conjugate-gradient
 * method with AMG as preconditioner.
 * 
 * @code
 *       ReductionControl coarse_grid_solver_control(mg_data.coarse_solver.maxiter,
 *                                                   mg_data.coarse_solver.abstol,
 *                                                   mg_data.coarse_solver.reltol,
 *                                                   false,
 *                                                   false);
 *       SolverCG<VectorType> coarse_grid_solver(coarse_grid_solver_control);
 *   
 *       std::unique_ptr<MGCoarseGridBase<VectorType>> mg_coarse;
 *   
 *       TrilinosWrappers::PreconditionAMG                 precondition_amg;
 *       TrilinosWrappers::PreconditionAMG::AdditionalData amg_data;
 *       amg_data.smoother_sweeps = mg_data.coarse_solver.smoother_sweeps;
 *   #ifdef DEAL_II_TRILINOS_WITH_EPETRA
 *       amg_data.n_cycles = mg_data.coarse_solver.n_cycles;
 *   #endif
 *       amg_data.smoother_type = mg_data.coarse_solver.smoother_type.c_str();
 *   
 *       precondition_amg.initialize(mg_matrices[min_level]->get_system_matrix(),
 *                                   amg_data);
 *   
 *       mg_coarse =
 *         std::make_unique<MGCoarseGridIterativeSolver<VectorType,
 *                                                      SolverCG<VectorType>,
 *                                                      LevelMatrixType,
 *                                                      decltype(precondition_amg)>>(
 *           coarse_grid_solver, *mg_matrices[min_level], precondition_amg);
 *   
 * @endcode
 * 
 * Finally, we create the Multigrid object, convert it to a preconditioner,
 * and use it inside of a conjugate-gradient solver to solve the linear
 * system of equations.
 * 
 * @code
 *       Multigrid<VectorType> mg(
 *         mg_matrix, *mg_coarse, mg_transfer, mg_smoother, mg_smoother);
 *   
 *       PreconditionerType preconditioner(dof, mg, mg_transfer);
 *   
 *       SolverCG<VectorType>(solver_control)
 *         .solve(fine_matrix, dst, src, preconditioner);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-Hybridpolynomialgeometricglobalcoarseningmultigridpreconditioner"></a> 
 * <h4>Hybrid polynomial/geometric-global-coarsening multigrid preconditioner</h4>
 * 
 
 * 
 * The above function deals with the actual solution for a given sequence of
 * multigrid objects. This functions creates the actual multigrid levels, in
 * particular the operators, and the transfer operator as a
 * MGTransferMatrixFree object.
 * 
 * @code
 *     template <typename VectorType, typename OperatorType, int dim>
 *     void solve_with_gmg(SolverControl                    &solver_control,
 *                         const OperatorType               &system_matrix,
 *                         VectorType                       &dst,
 *                         const VectorType                 &src,
 *                         const MultigridParameters        &mg_data,
 *                         const hp::MappingCollection<dim> &mapping_collection,
 *                         const DoFHandler<dim>            &dof_handler,
 *                         const hp::QCollection<dim>       &quadrature_collection)
 *     {
 * @endcode
 * 
 * Create a DoFHandler and operator for each multigrid level,
 * as well as, create transfer operators. To be able to
 * set up the operators, we need a set of DoFHandler that we create
 * via global coarsening of p or h. For latter, we need also a sequence
 * of Triangulation objects that are obtained by
 * Triangulation::coarsen_global().
 *     
 
 * 
 * In case no h-transfer is requested, we provide an empty deleter for the
 * `emplace_back()` function, since the Triangulation of our DoFHandler is
 * an external field and its destructor is called somewhere else.
 * 
 * @code
 *       MGLevelObject<DoFHandler<dim>>                     dof_handlers;
 *       MGLevelObject<std::unique_ptr<OperatorType>>       operators;
 *       MGLevelObject<MGTwoLevelTransfer<dim, VectorType>> transfers;
 *   
 *       std::vector<std::shared_ptr<const Triangulation<dim>>>
 *         coarse_grid_triangulations;
 *       if (mg_data.transfer.perform_h_transfer)
 *         coarse_grid_triangulations =
 *           MGTransferGlobalCoarseningTools::create_geometric_coarsening_sequence(
 *             dof_handler.get_triangulation());
 *       else
 *         coarse_grid_triangulations.emplace_back(
 *           &(dof_handler.get_triangulation()), [](auto *) {});
 *   
 * @endcode
 * 
 * Determine the total number of levels for the multigrid operation and
 * allocate sufficient memory for all levels.
 * 
 * @code
 *       const unsigned int n_h_levels = coarse_grid_triangulations.size() - 1;
 *   
 *       const auto get_max_active_fe_degree = [&](const auto &dof_handler) {
 *         unsigned int max = 0;
 *   
 *         for (auto &cell : dof_handler.active_cell_iterators())
 *           if (cell->is_locally_owned())
 *             max =
 *               std::max(max, dof_handler.get_fe(cell->active_fe_index()).degree);
 *   
 *         return Utilities::MPI::max(max, MPI_COMM_WORLD);
 *       };
 *   
 *       const unsigned int n_p_levels =
 *         MGTransferGlobalCoarseningTools::create_polynomial_coarsening_sequence(
 *           get_max_active_fe_degree(dof_handler), mg_data.transfer.p_sequence)
 *           .size();
 *   
 *       std::map<unsigned int, unsigned int> fe_index_for_degree;
 *       for (unsigned int i = 0; i < dof_handler.get_fe_collection().size(); ++i)
 *         {
 *           const unsigned int degree = dof_handler.get_fe(i).degree;
 *           Assert(fe_index_for_degree.find(degree) == fe_index_for_degree.end(),
 *                  ExcMessage("FECollection does not contain unique degrees."));
 *           fe_index_for_degree[degree] = i;
 *         }
 *   
 *       unsigned int minlevel = 0;
 *       unsigned int maxlevel = n_h_levels + n_p_levels - 1;
 *   
 *       dof_handlers.resize(minlevel, maxlevel);
 *       operators.resize(minlevel, maxlevel);
 *       transfers.resize(minlevel, maxlevel);
 *   
 * @endcode
 * 
 * Loop from the minimum (coarsest) to the maximum (finest) level and set up
 * DoFHandler accordingly. We start with the h-levels, where we distribute
 * on increasingly finer meshes linear elements.
 * 
 * @code
 *       for (unsigned int l = 0; l < n_h_levels; ++l)
 *         {
 *           dof_handlers[l].reinit(*coarse_grid_triangulations[l]);
 *           dof_handlers[l].distribute_dofs(dof_handler.get_fe_collection());
 *         }
 *   
 * @endcode
 * 
 * After we reached the finest mesh, we will adjust the polynomial degrees
 * on each level. We reverse iterate over our data structure and start at
 * the finest mesh that contains all information about the active FE
 * indices. We then lower the polynomial degree of each cell level by level.
 * 
 * @code
 *       for (unsigned int i = 0, l = maxlevel; i < n_p_levels; ++i, --l)
 *         {
 *           dof_handlers[l].reinit(dof_handler.get_triangulation());
 *   
 *           if (l == maxlevel) // finest level
 *             {
 *               auto &dof_handler_mg = dof_handlers[l];
 *   
 *               auto cell_other = dof_handler.begin_active();
 *               for (auto &cell : dof_handler_mg.active_cell_iterators())
 *                 {
 *                   if (cell->is_locally_owned())
 *                     cell->set_active_fe_index(cell_other->active_fe_index());
 *                   ++cell_other;
 *                 }
 *             }
 *           else // coarse level
 *             {
 *               auto &dof_handler_fine   = dof_handlers[l + 1];
 *               auto &dof_handler_coarse = dof_handlers[l + 0];
 *   
 *               auto cell_other = dof_handler_fine.begin_active();
 *               for (auto &cell : dof_handler_coarse.active_cell_iterators())
 *                 {
 *                   if (cell->is_locally_owned())
 *                     {
 *                       const unsigned int next_degree =
 *                         MGTransferGlobalCoarseningTools::
 *                           create_next_polynomial_coarsening_degree(
 *                             cell_other->get_fe().degree,
 *                             mg_data.transfer.p_sequence);
 *                       Assert(fe_index_for_degree.find(next_degree) !=
 *                                fe_index_for_degree.end(),
 *                              ExcMessage("Next polynomial degree in sequence "
 *                                         "does not exist in FECollection."));
 *   
 *                       cell->set_active_fe_index(fe_index_for_degree[next_degree]);
 *                     }
 *                   ++cell_other;
 *                 }
 *             }
 *   
 *           dof_handlers[l].distribute_dofs(dof_handler.get_fe_collection());
 *         }
 *   
 * @endcode
 * 
 * Next, we will create all data structures additionally needed on each
 * multigrid level. This involves determining constraints with homogeneous
 * Dirichlet boundary conditions, and building the operator just like on the
 * active level.
 * 
 * @code
 *       MGLevelObject<AffineConstraints<typename VectorType::value_type>>
 *         constraints(minlevel, maxlevel);
 *   
 *       for (unsigned int level = minlevel; level <= maxlevel; ++level)
 *         {
 *           const auto &dof_handler = dof_handlers[level];
 *           auto       &constraint  = constraints[level];
 *   
 *           constraint.reinit(dof_handler.locally_owned_dofs(),
 *                             DoFTools::extract_locally_relevant_dofs(dof_handler));
 *   
 *           DoFTools::make_hanging_node_constraints(dof_handler, constraint);
 *           VectorTools::interpolate_boundary_values(mapping_collection,
 *                                                    dof_handler,
 *                                                    0,
 *                                                    Functions::ZeroFunction<dim>(),
 *                                                    constraint);
 *           constraint.close();
 *   
 *           VectorType dummy;
 *   
 *           operators[level] = std::make_unique<OperatorType>(mapping_collection,
 *                                                             dof_handler,
 *                                                             quadrature_collection,
 *                                                             constraint,
 *                                                             dummy);
 *         }
 *   
 * @endcode
 * 
 * Set up intergrid operators and collect transfer operators within a single
 * operator as needed by the Multigrid solver class.
 * 
 * @code
 *       for (unsigned int level = minlevel; level < maxlevel; ++level)
 *         transfers[level + 1].reinit(dof_handlers[level + 1],
 *                                     dof_handlers[level],
 *                                     constraints[level + 1],
 *                                     constraints[level]);
 *   
 *       MGTransferMatrixFree<dim, typename VectorType::value_type> transfer(
 *         transfers, [&](const auto l, auto &vec) {
 *           operators[l]->initialize_dof_vector(vec);
 *         });
 *   
 * @endcode
 * 
 * Finally, proceed to solve the problem with multigrid.
 * 
 * @code
 *       mg_solve(solver_control,
 *                dst,
 *                src,
 *                mg_data,
 *                dof_handler,
 *                system_matrix,
 *                operators,
 *                transfer);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-ThecodeLaplaceProblemcodeclasstemplate"></a> 
 * <h3>The <code>LaplaceProblem</code> class template</h3>
 * 
 
 * 
 * Now we will finally declare the main class of this program, which solves
 * the Laplace equation on subsequently refined function spaces. Its structure
 * will look familiar as it is similar to the main classes of @ref step_27 "step-27" and
 * @ref step_40 "step-40". There are basically just two additions:
 * - The SparseMatrix object that would hold the system matrix has been
 * replaced by an object of the LaplaceOperator class for the MatrixFree
 * formulation.
 * - An object of parallel::CellWeights, which will help us with load
 * balancing, has been added.
 * 
 * @code
 *     template <int dim>
 *     class LaplaceProblem
 *     {
 *     public:
 *       LaplaceProblem(const Parameters &parameters);
 *   
 *       void run();
 *   
 *     private:
 *       void initialize_grid();
 *       void setup_system();
 *       void print_diagnostics();
 *       void solve_system();
 *       void compute_indicators();
 *       void adapt_resolution();
 *       void output_results(const unsigned int cycle);
 *   
 *       MPI_Comm mpi_communicator;
 *   
 *       const Parameters prm;
 *   
 *       parallel::distributed::Triangulation<dim> triangulation;
 *       DoFHandler<dim>                           dof_handler;
 *   
 *       hp::MappingCollection<dim> mapping_collection;
 *       hp::FECollection<dim>      fe_collection;
 *       hp::QCollection<dim>       quadrature_collection;
 *       hp::QCollection<dim - 1>   face_quadrature_collection;
 *   
 *       IndexSet locally_owned_dofs;
 *       IndexSet locally_relevant_dofs;
 *   
 *       AffineConstraints<double> constraints;
 *   
 *       LaplaceOperator<dim, double>               laplace_operator;
 *       LinearAlgebra::distributed::Vector<double> locally_relevant_solution;
 *       LinearAlgebra::distributed::Vector<double> system_rhs;
 *   
 *       std::unique_ptr<FESeries::Legendre<dim>> legendre;
 *       parallel::CellWeights<dim>               cell_weights;
 *   
 *       Vector<float> estimated_error_per_cell;
 *       Vector<float> hp_decision_indicators;
 *   
 *       ConditionalOStream pcout;
 *       TimerOutput        computing_timer;
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-ThecodeLaplaceProblemcodeclassimplementation"></a> 
 * <h3>The <code>LaplaceProblem</code> class implementation</h3>
 * 
 
 * 
 * 
 * <a name="step_75-Constructor"></a> 
 * <h4>Constructor</h4>
 * 
 
 * 
 * The constructor starts with an initializer list that looks similar to the
 * one of @ref step_40 "step-40". We again prepare the ConditionalOStream object to allow
 * only the first process to output anything over the console, and initialize
 * the computing timer properly.
 * 
 * @code
 *     template <int dim>
 *     LaplaceProblem<dim>::LaplaceProblem(const Parameters &parameters)
 *       : mpi_communicator(MPI_COMM_WORLD)
 *       , prm(parameters)
 *       , triangulation(mpi_communicator)
 *       , dof_handler(triangulation)
 *       , pcout(std::cout,
 *               (Utilities::MPI::this_mpi_process(mpi_communicator) == 0))
 *       , computing_timer(mpi_communicator,
 *                         pcout,
 *                         TimerOutput::never,
 *                         TimerOutput::wall_times)
 *     {
 *       Assert(prm.min_h_level <= prm.max_h_level,
 *              ExcMessage(
 *                "Triangulation level limits have been incorrectly set up."));
 *       Assert(prm.min_p_degree <= prm.max_p_degree,
 *              ExcMessage("FECollection degrees have been incorrectly set up."));
 *   
 * @endcode
 * 
 * We need to prepare the data structures for the hp-functionality in the
 * actual body of the constructor, and create corresponding objects for
 * every degree in the specified range from the parameter struct. As we are
 * only dealing with non-distorted rectangular cells, a linear mapping
 * object is sufficient in this context.
 *     
 
 * 
 * In the Parameters struct, we provide ranges for levels on which the
 * function space is operating with a reasonable resolution. The multigrid
 * algorithm requires linear elements on the coarsest possible level. So we
 * start with the lowest polynomial degree and fill the collection with
 * consecutively higher degrees until the user-specified maximum is
 * reached.
 * 
 * @code
 *       mapping_collection.push_back(MappingQ1<dim>());
 *   
 *       for (unsigned int degree = 1; degree <= prm.max_p_degree; ++degree)
 *         {
 *           fe_collection.push_back(FE_Q<dim>(degree));
 *           quadrature_collection.push_back(QGauss<dim>(degree + 1));
 *           face_quadrature_collection.push_back(QGauss<dim - 1>(degree + 1));
 *         }
 *   
 * @endcode
 * 
 * As our FECollection contains more finite elements than we want to use for
 * the finite element approximation of our solution, we would like to limit
 * the range on which active FE indices can operate on. For this, the
 * FECollection class allows to register a hierarchy that determines the
 * succeeding and preceding finite element in case of of p-refinement and
 * p-coarsening, respectively. All functions in the hp::Refinement namespace
 * consult this hierarchy to determine future FE indices. We will register
 * such a hierarchy that only works on finite elements with polynomial
 * degrees in the proposed range <code>[min_p_degree, max_p_degree]</code>.
 * 
 * @code
 *       const unsigned int min_fe_index = prm.min_p_degree - 1;
 *       fe_collection.set_hierarchy(
 *         /*next_index=*/
 *         [](const typename hp::FECollection<dim> &fe_collection,
 *            const unsigned int                    fe_index) -> unsigned int {
 *           return ((fe_index + 1) < fe_collection.size()) ? fe_index + 1 :
 *                                                            fe_index;
 *         },
 *         /*previous_index=*/
 *         [min_fe_index](const typename hp::FECollection<dim> &,
 *                        const unsigned int fe_index) -> unsigned int {
 *           Assert(fe_index >= min_fe_index,
 *                  ExcMessage("Finite element is not part of hierarchy!"));
 *           return (fe_index > min_fe_index) ? fe_index - 1 : fe_index;
 *         });
 *   
 * @endcode
 * 
 * We initialize the FESeries::Legendre object in the default configuration
 * for smoothness estimation.
 * 
 * @code
 *       legendre = std::make_unique<FESeries::Legendre<dim>>(
 *         SmoothnessEstimator::Legendre::default_fe_series(fe_collection));
 *   
 * @endcode
 * 
 * The next part is going to be tricky. During execution of refinement, a
 * few hp-algorithms need to interfere with the actual refinement process on
 * the Triangulation object. We do this by connecting several functions to
 * Triangulation::Signals: signals will be called at different stages during
 * the actual refinement process and trigger all connected functions. We
 * require this functionality for load balancing and to limit the polynomial
 * degrees of neighboring cells.
 *     
 
 * 
 * For the former, we would like to assign a weight to every cell that is
 * proportional to the number of degrees of freedom of its future finite
 * element. The library offers a class parallel::CellWeights that allows to
 * easily attach individual weights at the right place during the refinement
 * process, i.e., after all refine and coarsen flags have been set correctly
 * for hp-adaptation and right before repartitioning for load balancing is
 * about to happen. Functions can be registered that will attach weights in
 * the form that @f$a (n_\text{dofs})^b@f$ with a provided pair of parameters
 * @f$(a,b)@f$. We register such a function in the following.
 *     
 
 * 
 * Since we are only using information about the number of degrees of
 * freedom per cell, which is a quantity unique to every finite element, we
 * can compute the weights in advance.
 *     
 
 * 
 * For load balancing, efficient solvers like the one we use should scale
 * linearly with the number of degrees of freedom owned. We set the
 * parameters for cell weighting correspondingly: A weighting factor of @f$1@f$
 * and an exponent of @f$1@f$ (see the definitions of the `weighting_factor` and
 * `weighting_exponent` above).
 * 
 * @code
 *       const typename parallel::CellWeights<dim>::WeightingFunction
 *         weighting_function = parallel::CellWeights<dim>::ndofs_weighting(
 *           {prm.weighting_factor, prm.weighting_exponent});
 *       const std::vector<unsigned int> precomputed_weights =
 *         parallel::CellWeights<dim>::precompute_weights(fe_collection,
 *                                                        weighting_function);
 *       cell_weights.reinit(dof_handler, precomputed_weights);
 *   
 * @endcode
 * 
 * In h-adaptive applications, we ensure a 2:1 mesh balance by limiting the
 * difference of refinement levels of neighboring cells to one. With the
 * second call in the following code snippet, we will ensure the same for
 * p-levels on neighboring cells: levels of future finite elements are not
 * allowed to differ by more than a specified difference. The function
 * hp::Refinement::limit_p_level_difference takes care of this, but needs to
 * be connected to a very specific signal in the parallel context. The issue
 * is that we need to know how the mesh will be actually refined to set
 * future FE indices accordingly. As we ask the p4est oracle to perform
 * refinement, we need to ensure that the Triangulation has been updated
 * with the adaptation flags of the oracle first. An instantiation of
 * parallel::distributed::TemporarilyMatchRefineFlags does exactly
 * that for the duration of its life. Thus, we will create an object of this
 * class right before limiting the p-level difference, and connect the
 * corresponding lambda function to the signal
 * Triangulation::Signals::post_p4est_refinement, which will be triggered
 * after the oracle got refined, but before the Triangulation is refined.
 * Furthermore, we specify that this function will be connected to the front
 * of the signal, to ensure that the modification is performed before any
 * other function connected to the same signal.
 * 
 * @code
 *       triangulation.signals.post_p4est_refinement.connect(
 *         [&, min_fe_index]() {
 *           const parallel::distributed::TemporarilyMatchRefineFlags<dim>
 *             refine_modifier(triangulation);
 *           hp::Refinement::limit_p_level_difference(dof_handler,
 *                                                    prm.max_p_level_difference,
 *                                                    /*contains=*/min_fe_index);
 *         },
 *         boost::signals2::at_front);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-LaplaceProbleminitialize_grid"></a> 
 * <h4>LaplaceProblem::initialize_grid</h4>
 * 
 
 * 
 * For a L-shaped domain, we could use the function GridGenerator::hyper_L()
 * as demonstrated in @ref step_50 "step-50". However in the 2d case, that particular
 * function removes the first quadrant, while we need the fourth quadrant
 * removed in our scenario. Thus, we will use a different function
 * GridGenerator::subdivided_hyper_L() which gives us more options to create
 * the mesh. Furthermore, we formulate that function in a way that it also
 * generates a 3d mesh: the 2d L-shaped domain will basically elongated by 1
 * in the positive z-direction.
 *   
 
 * 
 * We first pretend to build a GridGenerator::subdivided_hyper_rectangle().
 * The parameters that we need to provide are Point objects for the lower left
 * and top right corners, as well as the number of repetitions that the base
 * mesh will have in each direction. We provide them for the first two
 * dimensions and treat the higher third dimension separately.
 *   
 
 * 
 * To create a L-shaped domain, we need to remove the excess cells. For this,
 * we specify the <code>cells_to_remove</code> accordingly. We would like to
 * remove one cell in every cell from the negative direction, but remove one
 * from the positive x-direction.
 *   
 
 * 
 * On the coarse grid, we set the initial active FE indices and distribute the
 * degrees of freedom once. We do that in order to assign the hp::FECollection
 * to the DoFHandler, so that all cells know how many DoFs they are going to
 * have. This step is mandatory for the weighted load balancing algorithm,
 * which will be called implicitly in
 * parallel::distributed::Triangulation::refine_global().
 * 
 * @code
 *     template <int dim>
 *     void LaplaceProblem<dim>::initialize_grid()
 *     {
 *       TimerOutput::Scope t(computing_timer, "initialize grid");
 *   
 *       std::vector<unsigned int> repetitions(dim);
 *       Point<dim>                bottom_left, top_right;
 *       for (unsigned int d = 0; d < dim; ++d)
 *         if (d < 2)
 *           {
 *             repetitions[d] = 2;
 *             bottom_left[d] = -1.;
 *             top_right[d]   = 1.;
 *           }
 *         else
 *           {
 *             repetitions[d] = 1;
 *             bottom_left[d] = 0.;
 *             top_right[d]   = 1.;
 *           }
 *   
 *       std::vector<int> cells_to_remove(dim, 1);
 *       cells_to_remove[0] = -1;
 *   
 *       GridGenerator::subdivided_hyper_L(
 *         triangulation, repetitions, bottom_left, top_right, cells_to_remove);
 *   
 *       const unsigned int min_fe_index = prm.min_p_degree - 1;
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         if (cell->is_locally_owned())
 *           cell->set_active_fe_index(min_fe_index);
 *   
 *       dof_handler.distribute_dofs(fe_collection);
 *   
 *       triangulation.refine_global(prm.min_h_level);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-LaplaceProblemsetup_system"></a> 
 * <h4>LaplaceProblem::setup_system</h4>
 * 
 
 * 
 * This function looks exactly the same to the one of @ref step_40 "step-40", but you will
 * notice the absence of the system matrix as well as the scaffold that
 * surrounds it. Instead, we will initialize the MatrixFree formulation of the
 * <code>laplace_operator</code> here. For boundary conditions, we will use
 * the Solution class introduced earlier in this tutorial.
 * 
 * @code
 *     template <int dim>
 *     void LaplaceProblem<dim>::setup_system()
 *     {
 *       TimerOutput::Scope t(computing_timer, "setup system");
 *   
 *       dof_handler.distribute_dofs(fe_collection);
 *   
 *       locally_owned_dofs = dof_handler.locally_owned_dofs();
 *       locally_relevant_dofs =
 *         DoFTools::extract_locally_relevant_dofs(dof_handler);
 *   
 *       locally_relevant_solution.reinit(locally_owned_dofs,
 *                                        locally_relevant_dofs,
 *                                        mpi_communicator);
 *       system_rhs.reinit(locally_owned_dofs, mpi_communicator);
 *   
 *       constraints.clear();
 *       constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
 *       DoFTools::make_hanging_node_constraints(dof_handler, constraints);
 *       VectorTools::interpolate_boundary_values(
 *         mapping_collection, dof_handler, 0, Solution<dim>(), constraints);
 *       constraints.close();
 *   
 *       laplace_operator.reinit(mapping_collection,
 *                               dof_handler,
 *                               quadrature_collection,
 *                               constraints,
 *                               system_rhs);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-LaplaceProblemprint_diagnostics"></a> 
 * <h4>LaplaceProblem::print_diagnostics</h4>
 * 
 
 * 
 * This is a function that prints additional diagnostics about the equation
 * system and its partitioning. In addition to the usual global number of
 * active cells and degrees of freedom, we also output their local
 * equivalents. For a regulated output, we will communicate the local
 * quantities with a Utilities::MPI::gather operation to the first process
 * which will then output all information. Output of local quantities is
 * limited to the first 8 processes to avoid cluttering the terminal.
 *   
 
 * 
 * On all other processes, the containers for the collected data remain empty.
 * To ensure that we do not access invalid memory with the insertion operator
 * (`<<`) on these processes, we need to check that the containers are not
 * empty.
 *   
 
 * 
 * Furthermore, we would like to print the frequencies of the polynomial
 * degrees in the numerical discretization. Since this information is only
 * stored locally, we will count the finite elements on locally owned cells
 * and later communicate them via Utilities::MPI::sum.
 * 
 * @code
 *     template <int dim>
 *     void LaplaceProblem<dim>::print_diagnostics()
 *     {
 *       const unsigned int first_n_processes =
 *         std::min<unsigned int>(8,
 *                                Utilities::MPI::n_mpi_processes(mpi_communicator));
 *       const bool output_cropped =
 *         first_n_processes < Utilities::MPI::n_mpi_processes(mpi_communicator);
 *   
 *       {
 *         pcout << "   Number of active cells:       "
 *               << triangulation.n_global_active_cells() << std::endl
 *               << "     by partition:              ";
 *   
 *         const std::vector<unsigned int> n_active_cells_per_subdomain =
 *           Utilities::MPI::gather(mpi_communicator,
 *                                  triangulation.n_locally_owned_active_cells());
 *         for (unsigned int i = 0; i < first_n_processes; ++i)
 *           if (n_active_cells_per_subdomain.size() > 0)
 *             pcout << ' ' << n_active_cells_per_subdomain[i];
 *         if (output_cropped)
 *           pcout << " ...";
 *         pcout << std::endl;
 *       }
 *   
 *       {
 *         pcout << "   Number of degrees of freedom: " << dof_handler.n_dofs()
 *               << std::endl
 *               << "     by partition:              ";
 *   
 *         std::vector<types::global_dof_index> n_dofs_per_subdomain =
 *           Utilities::MPI::gather(mpi_communicator,
 *                                  dof_handler.n_locally_owned_dofs());
 *         for (unsigned int i = 0; i < first_n_processes; ++i)
 *           if (n_dofs_per_subdomain.size() > 0)
 *             pcout << ' ' << n_dofs_per_subdomain[i];
 *         if (output_cropped)
 *           pcout << " ...";
 *         pcout << std::endl;
 *       }
 *   
 *       {
 *         std::vector<types::global_dof_index> n_constraints_per_subdomain =
 *           Utilities::MPI::gather(mpi_communicator, constraints.n_constraints());
 *   
 *         pcout << "   Number of constraints:        "
 *               << std::accumulate(n_constraints_per_subdomain.begin(),
 *                                  n_constraints_per_subdomain.end(),
 *                                  types::global_dof_index(0))
 *               << std::endl
 *               << "     by partition:              ";
 *         for (unsigned int i = 0; i < first_n_processes; ++i)
 *           if (n_constraints_per_subdomain.size() > 0)
 *             pcout << ' ' << n_constraints_per_subdomain[i];
 *         if (output_cropped)
 *           pcout << " ...";
 *         pcout << std::endl;
 *       }
 *   
 *       {
 *         std::vector<unsigned int> n_fe_indices(fe_collection.size(), 0);
 *         for (const auto &cell : dof_handler.active_cell_iterators())
 *           if (cell->is_locally_owned())
 *             n_fe_indices[cell->active_fe_index()]++;
 *   
 *         Utilities::MPI::sum(n_fe_indices, mpi_communicator, n_fe_indices);
 *   
 *         pcout << "   Frequencies of poly. degrees:";
 *         for (unsigned int i = 0; i < fe_collection.size(); ++i)
 *           if (n_fe_indices[i] > 0)
 *             pcout << ' ' << fe_collection[i].degree << ':' << n_fe_indices[i];
 *         pcout << std::endl;
 *       }
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-LaplaceProblemsolve_system"></a> 
 * <h4>LaplaceProblem::solve_system</h4>
 * 
 
 * 
 * The scaffold around the solution is similar to the one of @ref step_40 "step-40". We
 * prepare a vector that matches the requirements of MatrixFree and collect
 * the locally-relevant degrees of freedoms we solved the equation system. The
 * solution happens with the function introduced earlier.
 * 
 * @code
 *     template <int dim>
 *     void LaplaceProblem<dim>::solve_system()
 *     {
 *       TimerOutput::Scope t(computing_timer, "solve system");
 *   
 *       LinearAlgebra::distributed::Vector<double> completely_distributed_solution;
 *       laplace_operator.initialize_dof_vector(completely_distributed_solution);
 *   
 *       SolverControl solver_control(system_rhs.size(),
 *                                    prm.tolerance_factor * system_rhs.l2_norm());
 *   
 *       solve_with_gmg(solver_control,
 *                      laplace_operator,
 *                      completely_distributed_solution,
 *                      system_rhs,
 *                      prm.mg_data,
 *                      mapping_collection,
 *                      dof_handler,
 *                      quadrature_collection);
 *   
 *       pcout << "   Solved in " << solver_control.last_step() << " iterations."
 *             << std::endl;
 *   
 *       constraints.distribute(completely_distributed_solution);
 *   
 *       locally_relevant_solution.copy_locally_owned_data_from(
 *         completely_distributed_solution);
 *       locally_relevant_solution.update_ghost_values();
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-LaplaceProblemcompute_indicators"></a> 
 * <h4>LaplaceProblem::compute_indicators</h4>
 * 
 
 * 
 * This function contains only a part of the typical <code>refine_grid</code>
 * function from other tutorials and is new in that sense. Here, we will only
 * calculate all indicators for adaptation with actually refining the grid. We
 * do this for the purpose of writing all indicators to the file system, so we
 * store them for later.
 *   
 
 * 
 * Since we are dealing the an elliptic problem, we will make use of the
 * KellyErrorEstimator again, but with a slight difference. Modifying the
 * scaling factor of the underlying face integrals to be dependent on the
 * actual polynomial degree of the neighboring elements is favorable in
 * hp-adaptive applications @cite davydov2017hp. We can do this by specifying
 * the very last parameter from the additional ones you notices. The others
 * are actually just the defaults.
 *   
 
 * 
 * For the purpose of hp-adaptation, we will calculate smoothness estimates
 * with the strategy presented in the tutorial introduction and use the
 * implementation in SmoothnessEstimator::Legendre. In the Parameters struct,
 * we set the minimal polynomial degree to 2 as it seems that the smoothness
 * estimation algorithms have trouble with linear elements.
 * 
 * @code
 *     template <int dim>
 *     void LaplaceProblem<dim>::compute_indicators()
 *     {
 *       TimerOutput::Scope t(computing_timer, "compute indicators");
 *   
 *       estimated_error_per_cell.grow_or_shrink(triangulation.n_active_cells());
 *       KellyErrorEstimator<dim>::estimate(
 *         dof_handler,
 *         face_quadrature_collection,
 *         std::map<types::boundary_id, const Function<dim> *>(),
 *         locally_relevant_solution,
 *         estimated_error_per_cell,
 *         /*component_mask=*/ComponentMask(),
 *         /*coefficients=*/nullptr,
 *         /*n_threads=*/numbers::invalid_unsigned_int,
 *         /*subdomain_id=*/numbers::invalid_subdomain_id,
 *         /*material_id=*/numbers::invalid_material_id,
 *         /*strategy=*/
 *         KellyErrorEstimator<dim>::Strategy::face_diameter_over_twice_max_degree);
 *   
 *       hp_decision_indicators.grow_or_shrink(triangulation.n_active_cells());
 *       SmoothnessEstimator::Legendre::coefficient_decay(*legendre,
 *                                                        dof_handler,
 *                                                        locally_relevant_solution,
 *                                                        hp_decision_indicators);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-LaplaceProblemadapt_resolution"></a> 
 * <h4>LaplaceProblem::adapt_resolution</h4>
 * 
 
 * 
 * With the previously calculated indicators, we will finally flag all cells
 * for adaptation and also execute refinement in this function. As in previous
 * tutorials, we will use the "fixed number" strategy, but now for
 * hp-adaptation.
 * 
 * @code
 *     template <int dim>
 *     void LaplaceProblem<dim>::adapt_resolution()
 *     {
 *       TimerOutput::Scope t(computing_timer, "adapt resolution");
 *   
 * @endcode
 * 
 * First, we will set refine and coarsen flags based on the error estimates
 * on each cell. There is nothing new here.
 *     
 
 * 
 * We will use general refine and coarsen fractions that have been
 * elaborated in the other deal.II tutorials: using the fixed number
 * strategy, we will flag 30% of all cells for refinement and 3% for
 * coarsening, as provided in the Parameters struct.
 * 
 * @code
 *       parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number(
 *         triangulation,
 *         estimated_error_per_cell,
 *         prm.refine_fraction,
 *         prm.coarsen_fraction);
 *   
 * @endcode
 * 
 * Next, we will make all adjustments for hp-adaptation. We want to refine
 * and coarsen those cells flagged in the previous step, but need to decide
 * if we would like to do it by adjusting the grid resolution or the
 * polynomial degree.
 *     
 
 * 
 * The next function call sets future FE indices according to the previously
 * calculated smoothness indicators as p-adaptation indicators. These
 * indices will only be set on those cells that have refine or coarsen flags
 * assigned.
 *     
 
 * 
 * For the p-adaptation fractions, we will take an educated guess. Since we
 * only expect a single singularity in our scenario, i.e., in the origin of
 * the domain, and a smooth solution anywhere else, we would like to
 * strongly prefer to use p-adaptation over h-adaptation. This reflects in
 * our choice of a fraction of 90% for both p-refinement and p-coarsening.
 * 
 * @code
 *       hp::Refinement::p_adaptivity_fixed_number(dof_handler,
 *                                                 hp_decision_indicators,
 *                                                 prm.p_refine_fraction,
 *                                                 prm.p_coarsen_fraction);
 *   
 * @endcode
 * 
 * After setting all indicators, we will remove those that exceed the
 * specified limits of the provided level ranges in the Parameters struct.
 * This limitation naturally arises for p-adaptation as the number of
 * supplied finite elements is limited. In addition, we registered a custom
 * hierarchy for p-adaptation in the constructor. Now, we need to do this
 * manually in the h-adaptive context like in @ref step_31 "step-31".
 *     
 
 * 
 * We will iterate over all cells on the designated min and max levels and
 * remove the corresponding flags. As an alternative, we could also flag
 * these cells for p-adaptation by setting future FE indices accordingly
 * instead of simply clearing the refine and coarsen flags.
 * 
 * @code
 *       Assert(triangulation.n_levels() >= prm.min_h_level + 1 &&
 *                triangulation.n_levels() <= prm.max_h_level + 1,
 *              ExcInternalError());
 *   
 *       if (triangulation.n_levels() > prm.max_h_level)
 *         for (const auto &cell :
 *              triangulation.active_cell_iterators_on_level(prm.max_h_level))
 *           cell->clear_refine_flag();
 *   
 *       for (const auto &cell :
 *            triangulation.active_cell_iterators_on_level(prm.min_h_level))
 *         cell->clear_coarsen_flag();
 *   
 * @endcode
 * 
 * At this stage, we have both the future FE indices and the classic refine
 * and coarsen flags set. The latter will be interpreted by
 * Triangulation::execute_coarsening_and_refinement() for h-adaptation, and
 * our previous modification ensures that the resulting Triangulation stays
 * within the specified level range.
 *     
 
 * 
 * Now, we would like to only impose one type of adaptation on cells, which
 * is what the next function will sort out for us. In short, on cells which
 * have both types of indicators assigned, we will favor the p-adaptation
 * one and remove the h-adaptation one.
 * 
 * @code
 *       hp::Refinement::choose_p_over_h(dof_handler);
 *   
 * @endcode
 * 
 * In the end, we are left to execute coarsening and refinement. Here, not
 * only the grid will be updated, but also all previous future FE indices
 * will become active.
 *     
 
 * 
 * Remember that we have attached functions to triangulation signals in the
 * constructor, will be triggered in this function call. So there is even
 * more happening: weighted repartitioning will be performed to ensure load
 * balancing, as well as we will limit the difference of p-levels between
 * neighboring cells.
 * 
 * @code
 *       triangulation.execute_coarsening_and_refinement();
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-LaplaceProblemoutput_results"></a> 
 * <h4>LaplaceProblem::output_results</h4>
 * 
 
 * 
 * Writing results to the file system in parallel applications works exactly
 * like in @ref step_40 "step-40". In addition to the data containers that we prepared
 * throughout the tutorial, we would also like to write out the polynomial
 * degree of each finite element on the grid as well as the subdomain each
 * cell belongs to. We prepare necessary containers for this in the scope of
 * this function.
 * 
 * @code
 *     template <int dim>
 *     void LaplaceProblem<dim>::output_results(const unsigned int cycle)
 *     {
 *       TimerOutput::Scope t(computing_timer, "output results");
 *   
 *       Vector<float> fe_degrees(triangulation.n_active_cells());
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         if (cell->is_locally_owned())
 *           fe_degrees(cell->active_cell_index()) = cell->get_fe().degree;
 *   
 *       Vector<float> subdomain(triangulation.n_active_cells());
 *       for (auto &subd : subdomain)
 *         subd = triangulation.locally_owned_subdomain();
 *   
 *       DataOut<dim> data_out;
 *       data_out.attach_dof_handler(dof_handler);
 *       data_out.add_data_vector(locally_relevant_solution, "solution");
 *       data_out.add_data_vector(fe_degrees, "fe_degree");
 *       data_out.add_data_vector(subdomain, "subdomain");
 *       data_out.add_data_vector(estimated_error_per_cell, "error");
 *       data_out.add_data_vector(hp_decision_indicators, "hp_indicator");
 *       data_out.build_patches(mapping_collection);
 *   
 *       data_out.write_vtu_with_pvtu_record(
 *         "./", "solution", cycle, mpi_communicator, 2, 1);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-LaplaceProblemrun"></a> 
 * <h4>LaplaceProblem::run</h4>
 * 
 
 * 
 * The actual run function again looks very familiar to @ref step_40 "step-40". The only
 * addition is the bracketed section that precedes the actual cycle loop.
 * Here, we will pre-calculate the Legendre transformation matrices. In
 * general, these will be calculated on the fly via lazy allocation whenever a
 * certain matrix is needed. For timing purposes however, we would like to
 * calculate them all at once before the actual time measurement begins. We
 * will thus designate their calculation to their own scope.
 * 
 * @code
 *     template <int dim>
 *     void LaplaceProblem<dim>::run()
 *     {
 *       pcout << "Running with Trilinos on "
 *             << Utilities::MPI::n_mpi_processes(mpi_communicator)
 *             << " MPI rank(s)..." << std::endl;
 *   
 *       {
 *         pcout << "Calculating transformation matrices..." << std::endl;
 *         TimerOutput::Scope t(computing_timer, "calculate transformation");
 *         legendre->precalculate_all_transformation_matrices();
 *       }
 *   
 *       for (unsigned int cycle = 0; cycle < prm.n_cycles; ++cycle)
 *         {
 *           pcout << "Cycle " << cycle << ':' << std::endl;
 *   
 *           if (cycle == 0)
 *             initialize_grid();
 *           else
 *             adapt_resolution();
 *   
 *           setup_system();
 *   
 *           print_diagnostics();
 *   
 *           solve_system();
 *   
 *           compute_indicators();
 *   
 *           if (Utilities::MPI::n_mpi_processes(mpi_communicator) <= 32)
 *             output_results(cycle);
 *   
 *           computing_timer.print_summary();
 *           computing_timer.reset();
 *   
 *           pcout << std::endl;
 *         }
 *     }
 *   } // namespace Step75
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_75-main"></a> 
 * <h4>main()</h4>
 * 
 
 * 
 * The final function is the <code>main</code> function that will ultimately
 * create and run a LaplaceOperator instantiation. Its structure is similar to
 * most other tutorial programs.
 * 
 * @code
 *   int main(int argc, char *argv[])
 *   {
 *     try
 *       {
 *         using namespace dealii;
 *         using namespace Step75;
 *   
 *         Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
 *   
 *         Parameters        prm;
 *         LaplaceProblem<2> laplace_problem(prm);
 *         laplace_problem.run();
 *       }
 *     catch (std::exception &exc)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Exception on processing: " << std::endl
 *                   << exc.what() << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *   
 *         return 1;
 *       }
 *     catch (...)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Unknown exception!" << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         return 1;
 *       }
 *   
 *     return 0;
 *   }
 * @endcode
<a name="step_75-Results"></a><h1>Results</h1>
 
 
When you run the program with the given parameters on four processes in
release mode, your terminal output should look like this:
@code
Running with Trilinos on 4 MPI rank(s)...
Calculating transformation matrices...
Cycle 0:
   Number of active cells:       3072
     by partition:               768 768 768 768
   Number of degrees of freedom: 12545
     by partition:               3201 3104 3136 3104
   Number of constraints:        542
     by partition:               165 74 138 165
   Frequencies of poly. degrees: 2:3072
   Solved in 7 iterations.
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |     0.172s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| calculate transformation        |         1 |    0.0194s |        11% |
| compute indicators              |         1 |   0.00676s |       3.9% |
| initialize grid                 |         1 |     0.011s |       6.4% |
| output results                  |         1 |    0.0343s |        20% |
| setup system                    |         1 |   0.00839s |       4.9% |
| solve system                    |         1 |    0.0896s |        52% |
+---------------------------------+-----------+------------+------------+
 
 
Cycle 1:
   Number of active cells:       3351
     by partition:               875 761 843 872
   Number of degrees of freedom: 18228
     by partition:               4535 4735 4543 4415
   Number of constraints:        1202
     by partition:               303 290 326 283
   Frequencies of poly. degrees: 2:2522 3:829
   Solved in 7 iterations.
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |     0.165s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| adapt resolution                |         1 |   0.00473s |       2.9% |
| compute indicators              |         1 |   0.00764s |       4.6% |
| output results                  |         1 |    0.0243s |        15% |
| setup system                    |         1 |   0.00662s |         4% |
| solve system                    |         1 |     0.121s |        74% |
+---------------------------------+-----------+------------+------------+
 
 
...
 
 
Cycle 7:
   Number of active cells:       5610
     by partition:               1324 1483 1482 1321
   Number of degrees of freedom: 82047
     by partition:               21098 19960 20111 20878
   Number of constraints:        14357
     by partition:               3807 3229 3554 3767
   Frequencies of poly. degrees: 2:1126 3:1289 4:2725 5:465 6:5
   Solved in 7 iterations.
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |      1.83s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| adapt resolution                |         1 |   0.00834s |      0.46% |
| compute indicators              |         1 |    0.0178s |      0.97% |
| output results                  |         1 |    0.0434s |       2.4% |
| setup system                    |         1 |    0.0165s |       0.9% |
| solve system                    |         1 |      1.74s |        95% |
+---------------------------------+-----------+------------+------------+
@endcode
 
When running the code with more processes, you will notice slight
differences in the number of active cells and degrees of freedom. This
is due to the fact that solver and preconditioner depend on the
partitioning of the problem, which might yield to slight differences of
the solution in the last digits and ultimately yields to different
adaptation behavior.
 
Furthermore, the number of iterations for the solver stays about the
same in all cycles despite hp-adaptation, indicating the robustness of
the proposed algorithms and promising good scalability for even larger
problem sizes and on more processes.
 
Let us have a look at the graphical output of the program. After all
refinement cycles in the given parameter configuration, the actual
discretized function space looks like the following with its
partitioning on twelve processes on the left and the polynomial degrees
of finite elements on the right. In the left picture, each color
represents a unique subdomain. In the right picture, the lightest color
corresponds to the polynomial degree two and the darkest one corresponds
to degree six:
 
<div class="twocolumn" style="width: 80%; text-align: center;">
  <div>
    <img src="https://dealii.org/images/steps/developer/step-75.subdomains-07.svg"
         alt="Partitioning after seven refinements.">
  </div>
  <div>
    <img src="https://dealii.org/images/steps/developer/step-75.fedegrees-07.svg"
         alt="Local approximation degrees after seven refinements.">
  </div>
</div>
 
 
 
<a name="step-75-extensions"></a>
<a name="step_75-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
 
 
This tutorial shows only one particular way how to use parallel
hp-adaptive finite element methods. In the following paragraphs, you
will get to know which alternatives are possible. Most of these
extensions are already part of https://github.com/marcfehling/hpbox/,
which provides you with implementation examples that you can play
around with.
 
 
<a name="step_75-Differenthpdecisionstrategies"></a><h4>Different hp-decision strategies</h4>
 
 
The deal.II library offers multiple strategies to decide which type of
adaptation to impose on cells: either adjust the grid resolution or
change the polynomial degree. We only presented the <i>Legendre
coefficient decay</i> strategy in this tutorial, while @ref step_27 "step-27"
demonstrated the <i>Fourier</i> equivalent of the same idea.
 
See the "possibilities for extensions" section of @ref step_27 "step-27" for an
overview over these strategies, or the corresponding documentation
for a detailed description.
 
There, another strategy is mentioned that has not been shown in any
tutorial so far: the strategy based on <i>refinement history</i>. The
usage of this method for parallel distributed applications is more
tricky than the others, so we will highlight the challenges that come
along with it. We need information about the final state of refinement
flags, and we need to transfer the solution across refined meshes. For
the former, we need to attach the hp::Refinement::predict_error()
function to the Triangulation::Signals::post_p4est_refinement signal in
a way that it will be called <i>after</i> the
hp::Refinement::limit_p_level_difference() function. At this stage, all
refinement flags and future FE indices are terminally set and a reliable
prediction of the error is possible. The predicted error then needs to
be transferred across refined meshes with the aid of
parallel::distributed::CellDataTransfer.
 
Try implementing one of these strategies into this tutorial and observe
the subtle changes to the results. You will notice that all strategies
are capable of identifying the singularities near the reentrant corners
and will perform @f$h@f$-refinement in these regions, while preferring
@f$p@f$-refinement in the bulk domain. A detailed comparison of these
strategies is presented in @cite fehling2020 .
 
 
<a name="step_75-Solvewithmatrixbasedmethods"></a><h4>Solve with matrix-based methods</h4>
 
 
This tutorial focuses solely on matrix-free strategies. All hp-adaptive
algorithms however also work with matrix-based approaches in the
parallel distributed context.
 
To create a system matrix, you can either use the
LaplaceOperator::get_system_matrix() function, or use an
<code>assemble_system()</code> function similar to the one of @ref step_27 "step-27".
You can then pass the system matrix to the solver as usual.
 
You can time the results of both matrix-based and matrix-free
implementations, quantify the speed-up, and convince yourself which
variant is faster.
 
 
<a name="step_75-Multigridvariants"></a><h4>Multigrid variants</h4>
 
 
For sake of simplicity, we have restricted ourselves to a single type of
coarse-grid solver (CG with AMG), smoother (Chebyshev smoother with
point Jacobi preconditioner), and geometric-coarsening scheme (global
coarsening) within the multigrid algorithm. Feel free to try out
alternatives and investigate their performance and robustness.
 *
 *
<a name="step_75-PlainProg"></a>
<h1> The plain program</h1>
@include "step-75.cc"
*/